1. The problem is about understanding the concept of piecewise functions and their continuity. 2. A piecewise function is defined by different expressions depending on the input value's interval. 3. To check continuity at a point where the function's definition changes, we use the rule: $$\lim_{x \to c^-} f(x) = f(c) = \lim_{x \to c^+} f(x)$$ where $c$ is the point of change. 4. This means the left-hand limit, the function value at $c$, and the right-hand limit must all be equal for continuity. 5. For example, if $$f(x) = \begin{cases} x^2 & x < 1 \\ 2x + 1 & x \geq 1 \end{cases}$$ 6. We check continuity at $x=1$: 7. Calculate left limit: $$\lim_{x \to 1^-} x^2 = 1^2 = 1$$ 8. Calculate right limit: $$\lim_{x \to 1^+} 2x + 1 = 2(1) + 1 = 3$$ 9. Calculate function value: $$f(1) = 2(1) + 1 = 3$$ 10. Since left limit $1 \neq$ right limit $3$, the function is not continuous at $x=1$. This process applies to any piecewise function to determine continuity.